Neutron scattering from coherent admixtures of phonons with libronic excitations of diatomic impurities

The coherent mixed-mode states are represented in an $S$-matrix formalism. In the small-$\overrightarrow{\mathrm{q}}$ quasicontinuum region, the phonon field is expanded in terms of vector cubic harmonics, analogues of vector spherical harmonics. Tunneling states of the diatoms in a cubic crystal field are analyzed via group-theoretically projected tight-binding local oscillator states developed by Dick. Plausible level schemes are constructed by joining the strong-coupling spectrum they describe to the weak-coupling (free-rotator) spectrum. This semiphenomenological approach does not depend upon details of the generalized Devonshire hindering potential. It is found that the $T$-matrix elements connecting the phonons with the libronic or hindered-rotator excitations vanish in lowest nontrivial order unless the latter are of $E_g$ or $T_{2g}$ symmetry. Expressions for estimating the relative strengths of these allowed $T$-matrix elements are derived. Results are applied to neutron scattering experiments on dilute-impurity KCl: C${\mathrm{N}}^{-}$ and KBr: C${\mathrm{N}}^{-}$ and comparisons made with optical experiments on these systems. Of the usually considered $〈100〉$ and $〈111〉$ C${\mathrm{N}}^{-}$ impurity models for these crystals, our analysis shows that the $〈100〉$ model is eliminated by the neutron data, in agreement with earlier conclusions based upon optical and other experiments. The absence of a $T_{2g}$ interaction as observed in the neutron experiments presents a difficulty for the $〈111〉$ model if the libronically excited observed $E_g$ and unobserved $T_{2g}$ levels are split by an amount of the order of the ground-state tunnel splitting, i.e., $O\left(1\mathrm{} {\mathrm{cm}}^{-1\mathrm{}}\right)$. Under this assumption, the same difficulty exists for the $〈110〉$ model which we are also led to consider. The relative strengths of the $T_{2g}$ to $E_g$ couplings of the phonon to the libronic excitations are estimated for both the $〈111〉$ and $〈110〉$ models. Whereas the estimation procedure is perhaps too crude to distinguish between these two models on the basis of the neutron data alone, the fact that for either model the computed ratio of $T_{2g}$ to $E_g$ couplings is nearly equal to unity, under the assumption that the excited tunnel splittings are small, clearly suffices to rule out this latter assumption if either model is correct. It is concluded that the $T_{2g}$ rotational excitations of the C${\mathrm{N}}^{-}$ ions lie considerably above the observed $E_g$ libronic excitations and that the former are lifetime broadened to the extent that formation of coherent mixed modes does not occur; hence the absence of neutron observed splitting of the phonon modes in the $T_{2g}$ configuration. Our analysis of the Raman data lends support to this conclusion. Under the assumption of large excited-state splitting, our estimation procedure for the $\frac{T_{2g}}{E_g}$ coupling ratio does not apply and the neutron data are compatible with either the $〈111〉$ or $〈110〉$ models. The earlier experiments favor the $〈111〉$ model.